Two-Way Tables

Understanding Two-Way Tables

The following two-way table shows information about the hair colour of $100$ children:

What do the values mean?

Examples:

There are $textcolor{blue}{10}$ boys with black hair.

There are $textcolor{green}{18}$ children with blonde hair.

There are $textcolor{pink}{55}$ girls.

There are $textcolor{orange}{100}$ children overall.

Completing Two-Way Tables

You may be asked to complete incomplete two-way tables.

The following two-way table shows information about the hair colour of another $100$ children, with some missing values:

In this table, we need to find the missing values, $textcolor{red}{a}$ and $textcolor{blue}{b}$:

$textcolor{red}{a}$: we know there are $36$ children with brown hair, and $21$ of these are boys, so we can find how many girls have brown hair:

$36-21=15$

$textcolor{red}{a} = 15$

$textcolor{blue}{b}$: we know there are $100$ children overall, and $51$ of these are boys, so we can find how many girls there are overall:

$100-51=49$

$textcolor{blue}{b} = 49$

Finding Probabilities from Two-Way Tables

Two-way tables also allows us to calculate probabilities.

The following two-way table shows information about how $50$ people travel to school or work:

Examples of calculating probability:

• One person from this table is chosen at random. Calculate the probability of this person being under $25$ years old and walking to school or work.

From the table, we can see $11$ under $25$‘s walk to school or work, out of a possible $50$ people.

Therefore, the probability is $dfrac{11}{50}$.

• One person from this table is chosen at random. Calculate the probability of this person travelling by bus to school or work.

From the table, we can see $7$ people in total take the bus, out of a possible $50$ people.

Therefore, the probability is $dfrac{7}{50}$.

Conditional Probabilities from Two-Way Tables

We can also calculate conditional probabilities from two-way tables.

The following two-way table shows information about how $50$ people travel to school or work:

Conditional probability example:

• One person from this table is chosen at random. Given that the person chosen is over $25$, calculate the probability of this person driving in a car to school or work.

We are given that the person is over $25$, and as there are $25$ people over this age, this is the total number of possible people that can be chosen.

From the table, we can see $12$ over $25$‘s drive to school or work.

This gives us a probability of $12$ out of a total $25$ people:

$dfrac{12}{25}$